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According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed. So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.

What if we require that $C$ is fibrant over $A$?
If $C\to A$ is a (Serre) fibration, is $C$ exponentiable in the category of CG spaces above $A$?

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I doubt it. (-: – Mike Shulman Dec 5 '11 at 8:28

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